Chapter 8, Problem 24CR

To determine

Find the first six terms for the given sequence.

Answer to Problem 24CR

Thefirst six terms of the sequence are given below.

  $f\left(0\right)=4$

  $f\left(1\right)=6$

  $f\left(2\right)=10$

  $f\left(3\right)=16$

  $f\left(4\right)=24$

  $f\left(5\right)=34$

Explanation of Solution

Given:

The given datais $f\left(0\right)=4,f\left(n\right)=f\left(n-1\right)+2n$ .

Calculation:

The givensequenceis $f\left(n\right)=f\left(n-1\right)+2n$ .

The first term $f\left(0\right)=4$ .

For the second term

  $\begin{array}{l}f\left(1\right)=f\left(1-1\right)+2\cdot 1\\ \Rightarrow f\left(1\right)=f\left(0\right)+2\\ f\left(0\right)=4\\ \Rightarrow f\left(1\right)=4+2\\ \Rightarrow f\left(1\right)=6\end{array}$

For the third term

  $\begin{array}{l}f\left(2\right)=f\left(2-1\right)+2\cdot 2\\ \Rightarrow f\left(2\right)=f\left(1\right)+4\\ f\left(1\right)=6\\ \Rightarrow f\left(2\right)=6+4\\ \Rightarrow f\left(2\right)=10\end{array}$

For the fourth term

  $\begin{array}{l}f\left(3\right)=f\left(3-1\right)+2\cdot 3\\ \Rightarrow f\left(3\right)=f\left(2\right)+6\\ f\left(2\right)=10\\ \Rightarrow f\left(3\right)=10+6\\ \Rightarrow f\left(3\right)=16\end{array}$

For the fifth term

  $\begin{array}{l}f\left(4\right)=f\left(4-1\right)+2\cdot 4\\ \Rightarrow f\left(4\right)=f\left(3\right)+8\\ f\left(3\right)=16\\ \Rightarrow f\left(4\right)=16+8\\ \Rightarrow f\left(4\right)=24\end{array}$

For the sixth term

  $\begin{array}{l}f\left(5\right)=f\left(5-1\right)+2\cdot 5\\ \Rightarrow f\left(5\right)=f\left(4\right)+10\\ f\left(4\right)=24\\ \Rightarrow f\left(5\right)=24+10\\ \Rightarrow f\left(5\right)=34\end{array}$

Hence the first six terms of the sequence are given below.

  $f\left(0\right)=4$

  $f\left(1\right)=6$

  $f\left(2\right)=10$

  $f\left(3\right)=16$

  $f\left(4\right)=24$

  $f\left(5\right)=34$